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#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include "gr.h"
static char *xmalloc(int size)
{
char *result = (char *)malloc(size);
if (!result)
{
fprintf(stderr, "out of virtual memory\n");
abort();
}
return result;
}
/*!
* Cubic interpolation at position `x = xq` using four supporting points.
*
* \param[in] x Pointer to the supporting points' x-values
* \param[in] y Pointer to the supporting points' y-values
* \param[in] xq The x-coordinate to interpolate at
*
* \returns The resulting y-value
*/
static double cubic_interp(const double *x, const double *y, double xq)
{
double a[4], result;
a[0] = y[0];
a[1] = (y[1] - a[0]) / (x[1] - x[0]);
a[2] = (y[2] - a[0] - a[1] * (x[2] - x[0])) / ((x[2] - x[0]) * (x[2] - x[1]));
a[3] = (y[3] - a[0] - a[1] * (x[3] - x[0]) - a[2] * (x[3] - x[0]) * (x[3] - x[1])) /
((x[3] - x[0]) * (x[3] - x[1]) * (x[3] - x[2]));
/* Horner's method */
result = a[3];
result = result * (xq - x[2]) + a[2];
result = result * (xq - x[1]) + a[1];
result = result * (xq - x[0]) + a[0];
return result;
}
/*!
* Bilinear interpolation at position (`xq`, `yq`).
*
* \param[in] x Pointer to the supporting points' x-values
* \param[in] y Pointer to the supporting points' y-values
* \param[in] z Pointer to the supporting points' z-values
* \param[in] ix Index of next x-value less than `xq`
* \param[in] iy Index of next y-value less than `yq`
* \param[in] nx Number of given x-values
* \param[in] xq x-coordinate to interpolate at
* \param[in] yq y-coordinate to interpolate at
*
* \returns The resulting y-value
*/
static double bilinear_interp(const double *x, const double *y, const double *z, int ix, int iy, int nx, double xq,
double yq)
{
double f1, f2;
f1 = ((x[ix + 1] - xq) / (x[ix + 1] - x[ix])) * z[iy * nx + ix];
f1 += ((xq - x[ix]) / (x[ix + 1] - x[ix])) * z[iy * nx + ix + 1];
f2 = ((x[ix + 1] - xq) / (x[ix + 1] - x[ix])) * z[(iy + 1) * nx + ix];
f2 += ((xq - x[ix]) / (x[ix + 1] - x[ix])) * z[(iy + 1) * nx + ix + 1];
return ((y[iy + 1] - yq) / (y[iy + 1] - y[iy])) * f1 + ((yq - y[iy]) / (y[iy + 1] - y[iy])) * f2;
}
/*!
* Linear interpolation of the value at `x = xq`, using two supporting points.
*
* \param[in] x Pointer to left neighbour's x-value in supporting points,
* `x + 1` has to be a pointer to the left neighbour's x-value
* \param[in] y Pointer to left neighbour's y-value in supporting points,
* `y + 1` has to be a pointer to the left neighbour's y-value
* \param[in] xq The x-coordinate to interpolate at
*
* \returns The resulting y-value
*/
static double linear_interp(const double *x, const double *y, double xq)
{
return y[0] + ((y[1] - y[0]) / (x[1] - x[0])) * (xq - x[0]);
}
/*!
* Bicubic interpolation at (`xq`, `yq`). Uses linear interpolation as a
* fallback if cubic interpolation is not applicable.
*
* \param[in] x Pointer to the supporting points' x-values
* \param[in] y Pointer to the supporting points' y-values
* \param[in] z Pointer to the supporting points' z-values
* \param[in] ix Index of next x-value less than `xq`
* \param[in] iy Index of next y-value less than `yq`
* \param[in] nx Number of given x-values
* \param[in] ny Number of given y-values
* \param[in] xq The x-coordinate to interpolate at
* \param[in] yq The y-coordinate to interpolate at
*
* \returns The resulting z-value
*/
static double bicubic_interp(const double *x, const double *y, const double *z, int ix, int iy, int nx, int ny,
double xq, double yq)
{
double a[4];
if (ix + 2 >= nx || ix - 1 < 0)
{
if (iy + 2 >= ny || iy - 1 < 0)
{
/* fallback: linear interpolation */
return bilinear_interp(x, y, z, ix, iy, nx, xq, yq);
}
else
{
/* linear interpolation in X direction, cubic in Y direction */
a[0] = linear_interp(x + ix, z + ((iy - 1) * nx + ix), xq);
a[1] = linear_interp(x + ix, z + (iy * nx + ix), xq);
a[2] = linear_interp(x + ix, z + ((iy + 1) * nx + ix), xq);
a[3] = linear_interp(x + ix, z + ((iy + 2) * nx + ix), xq);
return cubic_interp(y + iy - 1, a, yq);
}
}
else
{
if (iy + 2 >= ny || iy - 1 < 0)
{
/* cubic interpolation in Y direction, linear in Y direction */
a[0] = cubic_interp(x + ix - 1, z + (iy * nx + ix - 1), xq);
a[1] = cubic_interp(x + ix - 1, z + ((iy + 1) * nx + ix - 1), xq);
return linear_interp(y + iy, a, yq);
}
else
{
/* cubic interpolation in both directions */
a[0] = cubic_interp(x + ix - 1, z + ((iy - 1) * nx + ix - 1), xq);
a[1] = cubic_interp(x + ix - 1, z + (iy * nx + ix - 1), xq);
a[2] = cubic_interp(x + ix - 1, z + ((iy + 1) * nx + ix - 1), xq);
a[3] = cubic_interp(x + ix - 1, z + ((iy + 2) * nx + ix - 1), xq);
return cubic_interp(y + iy - 1, a, yq);
}
}
}
/*!
* Creation of natural cubic splines
*
* \param[in] x Pointer to the x-values (nodes/supporting points)
* \param[in] y Pointer to the y-values (nodes/supporting points)
* \param[in] n Number of nodes/supporting points
* \param[out] spline Memory location of the `n * 4`
* target-array containing the splines
*/
static void create_splines(const double *x, const double *y, int n, double **spline)
{
int i;
double *h, *l, *m, *z, *alpha;
h = (double *)xmalloc((n - 1) * sizeof(double));
l = (double *)xmalloc(n * sizeof(double));
m = (double *)xmalloc((n - 1) * sizeof(double));
z = (double *)xmalloc(n * sizeof(double));
alpha = (double *)xmalloc((n - 1) * sizeof(double));
for (i = 0; i < n - 1; i++)
{
h[i] = x[i + 1] - x[i];
spline[i][0] = y[i];
}
spline[n - 1][0] = y[n - 1];
for (i = 1; i < n - 1; i++)
{
alpha[i] = (3. / h[i]) * (y[i + 1] - y[i]) - (3. / (h[i - 1])) * (y[i] - y[i - 1]);
}
l[0] = 1;
m[0] = 0;
z[0] = 0;
for (i = 1; i < n - 1; i++)
{
l[i] = 2 * (x[i + 1] - x[i - 1] - h[i - 1] * m[i - 1]);
m[i] = h[i] / l[i];
z[i] = (alpha[i] - h[i - 1] * z[i - 1]) / l[i];
}
l[n - 1] = 1;
z[n - 1] = 0;
spline[n - 1][2] = 0;
for (i = n - 2; i >= 0; i--)
{
spline[i][2] = z[i] - m[i] * spline[i + 1][2];
spline[i][1] = (spline[i + 1][0] - spline[i][0]) / h[i] - h[i] * ((spline[i + 1][2] + 2 * spline[i][2]) / 3);
spline[i][3] = (spline[i + 1][2] - spline[i][2]) / (3 * h[i]);
}
free(h);
free(l);
free(m);
free(z);
free(alpha);
}
/*!
* Interpolation in two dimensions using one of four different methods.
* The input points are located on a grid, described by `nx`, `ny`, `x`, `y` and `z`.
* The target grid ist described by `nxq`, `nyq`, `xq` and `yq` and the output
* is written to `zq` as a field of `nxq * nyq` values.
*
* \verbatim embed:rst:leading-asterisk
*
* The available methods for interpolation are the following:
*
* +-----------------+---+-------------------------------------------+
* | INTERP2_NEAREST | 0 | Nearest neighbour interpolation |
* +-----------------+---+-------------------------------------------+
* | INTERP2_LINEAR | 1 | Linear interpolation |
* +-----------------+---+-------------------------------------------+
* | INTERP_2_SPLINE | 2 | Interpolation using natural cubic splines |
* +-----------------+---+-------------------------------------------+
* | INTERP2_CUBIC | 3 | Cubic interpolation |
* +-----------------+---+-------------------------------------------+
*
* \endverbatim
*
* \param[in] nx The number of the input grid's x-values
* \param[in] ny The number of the input grid's y-values
* \param[in] x Pointer to the input grid's x-values
* \param[in] y Pointer to the input grid's y-values
* \param[in] z Pointer to the input grid's z-values (num. of values: nx * ny)
* \param[in] nxq The number of the target grid's x-values
* \param[in] nyq The number of the target grid's y-values
* \param[in] xq Pointer to the target grid's x-values
* \param[in] yq Pointer to the target grid's y-values
* \param[out] zq Pointer to the target grids's z-values, used for output
* \param[in] method Used method for interpolation
* \param[in] extrapval The extrapolation value
*/
void gr_interp2(int nx, int ny, const double *x, const double *y, const double *z, int nxq, int nyq, const double *xq,
const double *yq, double *zq, interp2_method_t method, double extrapval)
{
int ixq, iyq, ix, iy, ind, i;
double ***x_splines = NULL, **spline = NULL, *a = NULL, diff;
if (method == GR_INTERP2_SPLINE)
{
x_splines = (double ***)xmalloc(ny * sizeof(double **));
spline = (double **)xmalloc(ny * sizeof(double *));
for (ind = 0; ind < ny; ind++)
{
x_splines[ind] = (double **)xmalloc(ny * sizeof(double *));
for (i = 0; i < nx; i++)
{
x_splines[ind][i] = (double *)xmalloc(4 * sizeof(double));
}
spline[ind] = (double *)xmalloc(4 * sizeof(double));
create_splines(x, z + ind * nx, nx, x_splines[ind]);
}
a = (double *)xmalloc(ny * sizeof(double));
}
for (iyq = 0; iyq < nyq; iyq++)
{
for (ixq = 0; ixq < nxq; ixq++)
{
if ((xq[ixq] > x[nx - 1] || xq[ixq] < x[0]) || (yq[iyq] > y[ny - 1] || yq[iyq] < y[0]))
{
/* location outside of grid */
zq[iyq * nxq + ixq] = extrapval;
}
else
{
if (xq[ixq] > x[nx - 2])
{
/* index of next X value less than xq[ixq] is the second last */
ix = nx - 2;
}
else
{
ix = 0;
while (ix + 1 < nx && x[ix + 1] < xq[ixq])
{
/* searching for index of next X value less than xq[ixq] */
ix++;
}
}
if (yq[iyq] > y[ny - 2])
{
/* index of next Y value less than yq[iyq] is the second last */
iy = ny - 2;
}
else
{
iy = 0;
while (iy + 1 < ny && y[iy + 1] < yq[iyq])
{
/* searching for index of next Y value less than yq[iyq] */
iy++;
}
}
if (method == GR_INTERP2_NEAREST)
{
if (ix + 1 < nx && xq[ixq] - x[ix] > x[ix + 1] - xq[ixq])
{
ix++;
}
if (iy + 1 < ny && yq[iyq] - y[iy] > y[iy + 1] - yq[iyq])
{
iy++;
}
zq[iyq * nxq + ixq] = z[iy * nx + ix];
}
else if (method == GR_INTERP2_SPLINE)
{
/* interpolation in X direction: */
diff = xq[ixq] - x[ix];
for (ind = 0; ind < ny; ind++)
{
/* Horner's method */
a[ind] = x_splines[ind][ix][3];
a[ind] = a[ind] * diff + x_splines[ind][ix][2];
a[ind] = a[ind] * diff + x_splines[ind][ix][1];
a[ind] = a[ind] * diff + x_splines[ind][ix][0];
}
/* interpolation in Y direction: */
create_splines(y, a, ny, spline);
diff = yq[iyq] - y[iy];
/* Horner's method */
zq[iyq * nxq + ixq] = spline[iy][3];
zq[iyq * nxq + ixq] = zq[iyq * nxq + ixq] * diff + spline[iy][2];
zq[iyq * nxq + ixq] = zq[iyq * nxq + ixq] * diff + spline[iy][1];
zq[iyq * nxq + ixq] = zq[iyq * nxq + ixq] * diff + spline[iy][0];
}
else if (method == GR_INTERP2_CUBIC)
{
zq[iyq * nxq + ixq] = bicubic_interp(x, y, z, ix, iy, nx, ny, xq[ixq], yq[iyq]);
}
else if (method == GR_INTERP2_LINEAR)
{
zq[iyq * nxq + ixq] = bilinear_interp(x, y, z, ix, iy, nx, xq[ixq], yq[iyq]);
}
}
}
}
if (method == GR_INTERP2_SPLINE)
{
/* free allocated memory */
for (ind = 0; ind < ny; ind++)
{
for (i = 0; i < nx; i++)
{
free(x_splines[ind][i]);
}
free(x_splines[ind]);
free(spline[ind]);
}
free(x_splines);
free(spline);
free(a);
}
}
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